Chapter 14 - A Preview of Calculus: The Limit, Derivative, and Integral of a Function - 14.2 Algebra Techniques for Finding Limits - 14.2 Assess Your Understanding - Page 884: 49

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General formula for average rate of change from $a$ to $b$: $\frac{f(b)-f(a)}{b-a}.$ Factor each polynomial: $$\lim_{x\to 0}\frac{f(x)-f(0)}{x-0}\\=\lim_{x\to 0}\frac{3x^3-2x^2+4-(3(0)^3-2(0)^2+4)}{x}\\=\lim_{x\to 0}\frac{3x^3-2x^2}{x}\\=\lim_{x\to 0}\frac{x(3x^2-2x)}{x}.$$ Cancel the common factors: $$\lim_{x\to 0}(3x^2-2x)\\=3(0)^2-2(0)=0$$